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These notes, based on a graduate course I gave at Hamburg University in 2003, are intended to students having basic knowledges of differential geometry. Their main purpose is to provide a quick and accessible introduction to different…

Between 2017 and 2019, a standard Linear Algebra course from Instituto Superior T\'ecnico, University of Lisbon, used virtual learning content, mainly videos and formative assessment, delivered at the institution's MOOC platform to support…

These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…

This is a kind of introduction to some basic topics in analysis, some of which would be covered in standard graduate courses, and some not. However, an important difference is that not much in the way of prerequisites are needed, beyond…

The present article is an empirical study that investigates the learning situation of linear algebra. Research was performed among 60 science and engineering students from different universities in Zhejiang, Jiangsu, Hubei, and Shandong who…

The following is an exposition of a course of algebra that Prof. Aleksandr Aleksandrovich Zykov (1922-2013) distributed among the participants of his seminar in graph theory not far away from Odessa, Ukraine, on September, 1991. It is a…

This paper outlines our ideas on how to teach linear algebra in a mechanized mathematical environment, and discusses some of our reasons for thinking that this is a better way to teach linear algebra than the ``old fashioned way''. We…

These are introductory lecture notes on complex geometry, Calabi-Yau manifolds and toric geometry. We first define basic concepts of complex and Kahler geometry. We then proceed with an analysis of various definitions of Calabi-Yau…

The paper consists of two parts. The first part is devoted to logic for universal algebraic geometry. The second one deals with problems and some results. It may be regarded as a brief exposition of some ideas from the book in progress:…

These notes are for the author's lectures, "Integral Reduction and Applied Algebraic Geometry Techniques" in the School and Workshop on Amplitudes in Beijing 2016. I introduce the applications of algebraic geometry methods on multi-loop…

This book is expository and is in Russian (sample English translation of two pages is given). It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear different notions of…

In past few decades, tensor algebra also known as multi-linear algebra has been developed and customized as a tool to be used for various engineering applications. In particular, with the help of a special form of tensor contracted product,…

This article offers a motivating travel guide towards the Jordan normal form, one of the highlights in courses on linear algebra or advanced mathematics. Its itinerary is characterized by a focus on core geometric aspects and the avoidance…

In this work, the authors describe efforts aimed at Indigenizing a second-year linear algebra course at a small liberal arts university in Manitoba, Canada. This is done through an assignment, part hands-on and part written work, that…

These are lecture notes that are based on the lectures from a class I taught on the topic of Randomized Linear Algebra (RLA) at UC Berkeley during the Fall 2013 semester.

This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…

This is the abstract of a series of lectures given during the XIIIth School on Geometry and Physics, Bialystok (Poland), in July 2024. In this minicourse, we first examine the algebraic aspects of barycentric algebras. Then, we focus on…

These lecture notes provide an introduction to logarithmic geometry with a view towards recent applications in the desingularization theory.

This textbook is an introduction to economic networks, intended for students and researchers in the fields of economics and applied mathematics. The textbook emphasizes quantitative modeling, with the main underlying tools being graph…

Some notions of algebraic geometry can be defined for arbitrary varieties of algebras. This leads to universal algebraic geometry. The main idea of the presented theory is to consider interactions between algebra, logic and geometry in…